Reachable sets of second-order switched systems

This paper studies the reachability of second-order switched nonlinear systems in which the switching law is the only control action. For some initial conditions we find states that are reachable from the initial condition. We also give the switching law that drives the system from the initial state to the final state. Moreover, for switched systems with limit cycles, it is proved that the reachable set is totally determined for each switched convergent state. Finally, several examples and a real application are presented in order to illustrate the results.     

Reachability analysis of constrained switched linear systems

In this note, we investigate the reachability of switched linear systems with switching/input constraints. We prove that, under a mild assumption of the feasible switching signals, the reachability set is the reachable subspace of the unconstrained system. We also address the local reachability for switched linear systems with input constraints and present a complete criterion for a general class of switched linear systems.     

A constructive approach to reachability realization of discrete-time switched linear systems

This paper studies the reachability of discrete-time switched linear systems. A new constructive approach to designing switching sequences is developed so that not only the reachable subspace is realized but also the required number of switchings is reduced significantly. Detailed comparison between the obtained result and that reported in the open literature is also made, which verifies the advantage of the proposed approach.     

Controllability and reachability criteria for switched linear systems

This paper investigates the controllability and reachability of switched linear control systems. It is proven that both the controllable and reachable sets are subspaces of the total space. Complete geometric characterization for both sets is presented. The switching control design problem is also addressed.     

Reachability and observability of switched linear systems with continuous-time and discrete-time subsystems

In this paper, the reachability and observability criteria of switched linear systems with continuous-time and discrete-time subsystems are obtained. These criteria show that the reachable set may not be a subspace in the state space, because of the existence of discrete-time subsystems. Therefore, the definition of span reachability is proposed. Moreover, we demonstrate that the reachable set is equivalent to subspace if the discrete-time subsystems are reversible. The subspace algorithms for span reachability and unobservability are provided. One example is introduced to illustrate the effectiveness of the proposed criteria.     

Neural networks stabilization and disturbance attenuation for nonlinear switched impulsive systems

In this paper, we address the problem of neural networks (NNs) stabilization and disturbance rejection for a class of nonlinear switched impulsive systems. An adaptive NN feedback control scheme and an impulsive controller for output tracking error disturbance attenuation of nonlinear switched impulsive systems are given under all admissible switched strategy based on NN. The NN is used to compensate for the nonlinear uncertainties of switched impulsive systems, and the approximation error of NN is introduced to the adaptive law in order to improve the tracking attenuation quality of the switched impulsive systems. Impulsive controller is designed to attenuate effect of switching impulse. Under all admissible switching law, impulsive controller and adaptive NN feedback controller can guarantee asymptotic stability of tracking error and improve disturbance attenuation level of tracking error for the overall nonlinear switched impulsive system. Finally, a numerical example is given to demonstrate the effectiveness of the proposed control and stabilization methods.     

Periodic stabilizability of switched linear control systems

Stabilizability via direct/observer-based state feedback control for discrete-time switched linear control systems (SLCSs) is investigated in this paper. For an SLCS, the control factors are not only the control input but also the switching signal, and they need to be designed in order to stabilize the system. As a result, stabilization design for SLCSs is more complicated than that for non-switched ones. Differently from the existing approaches, a periodic switching signal and piecewise constant linear state feedback control are adopted to achieve periodic stabilizability for such systems. It is highlighted that multiple feedback controllers need to be designed for one subsystem. For discrete-time SLCSs, it is proved that reachability implies periodic stabilizability via state feedback. A necessary and sufficient criterion for periodic stabilizability is also established. Two stabilization design algorithms are presented for real application. Moreover, it is proved that reachability and observability imply periodic stabilizability via observer-based feedback for discrete-time SLCSs. Periodic detectability, as the dual concept of periodic stabilizability, is discussed and the stabilization design algorithms via observer-based feedback are presented as well.     

Stability analysis and control synthesis of switched impulsive systems

In this paper, we investigate the stability analysis problem of switched impulsive nonlinear systems and several stabilization problems of switched discrete‐time linear systems are studied. First, sufficient conditions ensuring globally uniformly asymptotically stability of switched nonlinear impulsive system under arbitrary and DDT (dynamical dwell time which defines the length of the time interval between two successive switchings) switching are derived, respectively. In the DDT switching case, we first consider the switched system composed by stable subsystems, then we extend the results to the case where not all subsystems are stable. The stabilizations of switched discrete‐time linear system under arbitrary switching, DDT switching and asynchronous switching are investigated respectively. Based on the stability analysis results, the control synthesis consists of controller design for each subsystem and state impulsive jumping generators design at switching instant. With the aid of the state impulsive jumping generators at switching instant, the ‘energy’ produced by switching can be minimized, which leads to less conservative results. Several numerical examples are given to illustrate the proposed results within this paper. Copyright © 2011 John Wiley & Sons, Ltd.     

Duality Properties of Linear Impulsive Systems

Reachability and observability properties of linear switched, impulsive, and switched/impulsive systems have received a great deal of attention over the past decade. The purpose of this paper is to rigorously establish a duality between reachability and observability for linear impulsive systems both in direct system‐theoretic terms as well as in geometric terms. These connections are established using an appropriately defined adjoint system associated with the original impulsive system. The duality between reachability and observability is then used to leverage recent reachability results in order to tighten the relationship between the set of unobservable states and a certain invariant subspace.     

Realization theory for linear switched systems: Formal power series approach

The paper deals with the realization theory of linear switched systems. Necessary and sufficient conditions are formulated for a family of input–output maps to be realizable by linear switched systems. Characterization of minimal realizations is presented. The paper treats two types of linear switched systems. The first one is when all switching sequences are allowed. The second one is when only a subset of switching sequences is admissible, but within this restricted set the switching times are arbitrary. The paper uses the theory of formal power series to derive the results on realization theory.     

On the reachability in any fixed time for positive continuous-time linear systems

This paper deals with the reachability of continuous-time linear positive systems. The reachability of such systems, which we will call here the strong reachability, amounts to the possibility of steering the state in any fixed time to any point of the positive orthant by using nonnegative control functions. The main result of this paper essentially says that the only strongly reachable positive systems are those made of decoupled scalar subsystems. Moreover, the strongly reachable set is also characterized.     

On Controllability of Switched Linear Systems

This paper investigates the number of switchings and design of switching sequences for controllability of switched linear systems. Two related results are established. One is a new constructive approach to designing switching sequences. The controllable state set of each switching sequence designed via the approach coincides with the controllable subspace of switched linear systems. The other is a well-estimated value for the minimum number of switchings required for controllability. Each state in the controllable subspace can be steered to origin within this value of switching times.     

Asynchronous switching output feedback control of discrete-time switched linear systems

In this paper, the problem of dynamic output-feedback control synthesis is addressed for discrete-time switched linear systems under asynchronous switching. The proposed hybrid controller consists of a standard dynamic output-feedback switching control law and an impulsive reset law induced by controller state jumps. Using the average dwell time technique incorporating with multiple quadratic Lyapunov functions, the switching control synthesis conditions for asymptotic stability with guaranteed weighted ?₂-gain performance are derived as a set of linear matrix inequalities (LMIs). The proposed hybrid synthesis scheme advances existing design methods for output-feedback asynchronous switching control of switched linear systems in two important aspects: LMI formulation of the synthesis problem; and arbitrary order of the controller state. A numerical example is used to illustrate the effectiveness and advantages of the proposed design technique.     

Reachability and controllability of switched linear discrete-timesystems

This paper investigates the reachability and controllability issues for switched linear discrete-time systems. Geometric characterization of controllability is presented. For reversible systems, the controllable sets and the reachable sets are identified in Wonham's geometric approach, and verifiable conditions for reachability and controllability are also presented     

Reachability and observability of linear impulsive systems

Linear impulsive systems constitute a class of hybrid systems in which the state propagates according to linear continuous-time dynamics except for a countable set of times at which the state can change instantaneously. While in general these impulsive effects can be time-driven and/or event-driven, here we focus our attention on the time-driven case. For this class of systems, we address the fundamental concepts of reachability and observability. In particular, we present a geometric characterization of the reachable and unobservable sets in terms of invariant subspaces and provide algorithms for their construction.     

Finite data-rate feedback stabilization of switched and hybrid linear systems

We study the problem of asymptotically stabilizing a switched linear control system using sampled and quantized measurements of its state. The switching is assumed to be slow enough in the sense of combined dwell time and average dwell time, each individual mode is assumed to be stabilizable, and the available data rate is assumed to be large enough but finite. Our encoding and control strategy is rooted in the one proposed in our earlier work on non-switched systems, and in particular the data-rate bound used here is the data-rate bound from that earlier work maximized over the individual modes. The main technical step that enables the extension to switched systems concerns propagating over-approximations of reachable sets through sampling intervals, during which the switching signal is not known; a novel algorithm is developed for this purpose. Our primary focus is on systems with time-dependent switching (switched systems) but the setting of state-dependent switching (hybrid systems) is also discussed.     

Robust stabilization of uncertain impulsive switched systems with delayed control

In this paper, stability criteria and switching controllers’ design problems for uncertain impulsive switched systems with input delay are investigated by using the receding horizon method. Some LMI conditions are derived to guarantee asymptotical stability of an impulsive switched system under a certain designed delayed controller. Finally, a numerical example is presented to illustrate the effectiveness of the results obtained.     

Asynchronous Switching Control for Continuous-time Switched Linear Systems with Output-feedback

This paper investigates the asynchronous switching control problem for continuous-time switched linear systems via dynamic output-feedback, where the dynamic output-feedback controller contains an impulsive reset law to reset the controller state. A time-varying multiple Lyapunov-like-function (MLF) approach is employed to analyze the stability and weighted L₂-gain of the closed-loop systems. The switching stability criteria for the closed-loop systems are established in terms of linear matrix inequalities (LMIs), which are dependent on the upper and lower bounds of the switching interval and the asynchronous delays. The switching logic is designed to guarantee the closed-loop systems achieving the weighted L₂-gain performance. Two numerical examples are provided to show the effectiveness of the proposed method.     

Measure-controlled dynamic systems: Polyhedral approximation of their reachable set boundary

An algorithm for polyhedral approximation of the reachable set of impulsive dynamic control systems is designed. The boundary points of the reachable set are determined by recursively generating and solving a family of auxiliary optimal impulsive control problems with state-linear objective functional. The impulsive control problem is solved with an algorithm that implicitly reduces the problem an ordinary optimal control problem. The reduced problem thus obtained is solved with an algorithm based on local approximations of the reachable set.     

Robust finite-time guaranteed cost control for impulsive switched systems with time-varying delay

This paper is concerned with the problem of robust finite-time guaranteed cost control for a class of impulsive switched systems with time-varying delay. Firstly, the definitions of finite-time boundedness, finite-time stabilization, and robust finite-time guaranteed cost control are presented. Next, based on the average dwell-time approach, sufficient conditions on robust finite-time guaranteed cost control are obtained for the uncertain impulsive switched systems. Then, by using multiple Lyapunov-like functional method and linear matrix equality (LMI) technique, the state feedback controller is designed to guarantee that the uncertain impulsive switched system is finite-time stabilized. Furthermore, a finite-time guaranteed cost bound is given. Finally, a numerical example is shown to illustrate the effectiveness of the proposed results.